Pulsar navigation may allow to determine the direction in which the spacecraft is traveling, but not the actual distance from Earth, unless the spacecraft has traveled very large distances.
A simple example. Last time I checked, Voyager1 was 18 light-hours away from Earth. Now Alpha Centauri is 4 light-years away. Multiply 4 by 365 and then by 24 and divide by 18.
That tells us that Alpha Centauri is about 2000 times more distant than Voyager1.
Put two dots on a blackboard, 2 meters apart. The dots represent Earth and Alpha Centauri. Then Voyager1, the spacecraft that has traveled farthest from Earth, is a full 1 millimeter away from the Earth point.
The closest pulsar to Earth is at an estimated distance of about 250 light-years. The pulsars will be very useful to find the direction, but for distance timing radio signals from Earth to the spacecraft remains the best method.